Standard-Deviation Normalization Proof

Rowan Brad Quni-Gudzinas

#Fundamental

Role of Standard-Deviation Normalization in a Scale-Invariant Statistical Framework

Author: Rowan Brad Quni-Gudzinas Affiliation: QNFO Contact: rowan.quni@qnfo.org ORCID: 0009-0002-4317-5604 ISNI: 0000 0005 2645 6062 DOI: Publication Date: 2025-09-28 Version: 1.0

#Given

Let $(\Omega, \mathcal{F}, P)$be a probability space, and let $X$be a real-valued random variable defined on this space, representing a physical observable. Assume $X$has a well-defined mean $\mu = \mathbb{E}[X]$and finite, positive variance $\sigma^2 = \mathbb{E}[(X - \mu)^2] \gt 0$.

Consider the standardized random variable:

Z := \frac{X - \mu}{\sigma}.

Let $\mathcal{M}$be a statistical manifold representing a family of probability distributions $\{p(x|\theta)\}_{\theta \in \Theta}$, where $\theta$are parameters. Assume the family includes location-scale distributions of the form $p(x|\mu, \sigma) = \sigma^{-1} f((x-\mu)/\sigma)$, where $f$is a fixed probability density.

#Goal

Demonstrate that standard-deviation normalization (i.e., measuring observables in units of their inherent variability $\sigma$) is fundamental to constructing a statistically unified, scale-invariant description of reality, as required by the Scale-Invariant Epistemic Framework.

#Derivation

#Step

1: Standard-Deviation Normalization Defines Scale-Invariant Coordinates

Define the dimensionless parameters:

\eta := \frac{\mu}{\sigma}, \quad \xi := \log \sigma. \tag{1}

These are the natural coordinates for the location-scale family. The original parameters are recovered via $\mu = e^\xi \eta$, $\sigma = e^\xi$.

The transformation $(\mu, \sigma) \mapsto (\eta, \xi)$is precisely the mathematical operation of standard-deviation normalization. The location parameter $\mu$is expressed in units of the scale parameter $\sigma$, and the scale parameter itself is expressed logarithmically.

#Step

2: The Standardized Variable Z is Invariant under Scaling

Consider a global scale transformation of the observable $X$:

X \mapsto X' = \lambda X, \quad \lambda \gt 0.

The mean and standard deviation transform as:

\mu' = \lambda \mu, \quad \sigma' = \lambda \sigma.

The standardized variable transforms as:

Z' = \frac{X' - \mu'}{\sigma'} = \frac{\lambda X - \lambda \mu}{\lambda \sigma} = \frac{X - \mu}{\sigma} = Z.

Thus, $Z$is invariant under global scaling of the original variable $X$.

#Step

3: The Fisher Metric is Scale-Invariant in Normalized Coordinates

For the location-scale family $p(x|\mu,\sigma) = \sigma^{-1} f((x-\mu)/\sigma)$, the standard Fisher metric components are:

g_{\mu\mu} = \frac{I_1}{\sigma^2}, \quad g_{\sigma\sigma} = \frac{I_2}{\sigma^2}, \quad g_{\mu\sigma} = \frac{I_3}{\sigma^2},

where $I_1, I_2, I_3$are constants depending only on the base density $f$.

Transforming to the normalized coordinates $(\eta, \xi)$using the Jacobian:

\frac{\partial(\mu, \sigma)}{\partial(\eta, \xi)} = \begin{pmatrix} e^\xi & 0 \\ e^\xi \eta & e^\xi \end{pmatrix},

the metric components become:

\tilde{g}_{\eta\eta} = g_{\mu\mu} (e^\xi)^2 + 2 g_{\mu\sigma} (e^\xi)(e^\xi \eta) + g_{\sigma\sigma} (e^\xi \eta)^2 = \frac{1}{\sigma^2}(I_1 + 2I_3\eta + I_2\eta^2) \cdot \sigma^2 = I_1 + 2I_3\eta + I_2\eta^2, \tag{2a}

\tilde{g}_{\xi\xi} = g_{\sigma\sigma} (e^\xi)^2 = \frac{I_2}{\sigma^2} \cdot \sigma^2 = I_2, \tag{2b}

\tilde{g}_{\eta\xi} = g_{\mu\sigma} (e^\xi)(e^\xi) + g_{\sigma\sigma} (e^\xi \eta)(e^\xi) = \frac{1}{\sigma^2}(I_3 + I_2\eta) \cdot \sigma^2 = I_3 + I_2\eta. \tag{2c}

All components $\tilde{g}_{ij}(\eta, \xi)$are independent of the scale parameter $\sigma$ (or $\xi$). The metric is therefore scale-invariant in the normalized coordinates $(\eta, \xi)$.

The line element is:

ds^2 = \tilde{g}_{ij}(\eta, \xi) \, d\theta^i d\theta^j = (I_1 + 2I_3\eta + I_2\eta^2) d\eta^2 + 2(I_3 + I_2\eta) d\eta d\xi + I_2 d\xi^2. \tag{3}

This metric is manifestly invariant under the transformation $\eta \mapsto \lambda \eta$, $\xi \mapsto \xi + \log \lambda$, which corresponds to the original scaling $X \mapsto \lambda X$.

#Step

4: Normalized Statistical Distances Are Scale-Invariant

The geodesic distance between two points $(\eta_1, \xi_1)$and $(\eta_2, \xi_2)$on the statistical manifold is:

d[(\eta_1, \xi_1), (\eta_2, \xi_2)] = \int_{\gamma} \sqrt{ds^2},

where $\gamma$is the geodesic path. Since the integrand $ds^2$is invariant under the scaling transformation, the distance $d$is also invariant.

This implies that the statistical distinguishability between two distributions, measured by their geodesic distance in the normalized parameter space, is independent of the absolute scale of the observables. It depends only on the dimensionless ratios $\eta$and the relative scale $\Delta \xi = \xi_2 - \xi_1$.

#Step

5: Connection to Physical Quantities and Information Theory

The standardized variable $Z = (X - \mu)/\sigma$represents the number of standard deviations $X$deviates from its mean. This is a dimensionless measure of statistical significance.

The Signal-to-Noise Ratio (SNR) is a fundamental quantity in information theory and physics:

\text{SNR} = \frac{\mu}{\sigma} = \eta.

This is precisely the normalized location parameter $\eta$.

The Coefficient of Variation (CV) is another scale-invariant measure:

\text{CV} = \frac{\sigma}{\mu} = \frac{1}{\eta}.

The Fisher information matrix, when expressed in normalized coordinates, yields dimensionless components $\tilde{g}_{ij}$, representing the information content per unit of standard deviation.

#Step 6:

Universality and Critical Phenomena

In the context of critical phenomena, the correlation function $\langle \mathcal{O}(0) \mathcal{O}(r) \rangle$is often normalized by the variance $\langle \mathcal{O}^2 \rangle$:

G(r) = \frac{\langle \mathcal{O}(0) \mathcal{O}(r) \rangle}{\langle \mathcal{O}^2 \rangle}.

This normalized function $G(r)$exhibits universal scaling behavior $G(r) \sim r^{-\eta}$near critical points, where $\eta$is a universal critical exponent (dimensionless). The normalization by variance ensures that $G(r)$is dimensionless and scale-invariant.

#Step

7: Implications for the Scale-Invariant Epistemic Framework

The scale invariance of the Fisher metric in normalized coordinates $(\eta, \xi)$demonstrates that:

Standard-deviation normalization provides the correct coordinate system for describing statistical manifolds in a scale-invariant manner.

Statistical distances and distinguishability measures become scale-invariant when computed using normalized parameters.

Physical laws expressed in terms of normalized quantities (like SNR, CV, correlation functions normalized by variance) automatically satisfy the principle of universal scale invariance.

The standard deviation $\sigma$acts as the natural unit for measuring the variability of an observable $X$, consistent with the epistemic principle that knowledge is limited by inherent fluctuations.

#Conclusion

Theorem (Fundamental Role of Standard-Deviation Normalization). Standard-deviation normalization, achieved by expressing observables as $Z = (X - \mu)/\sigma$and parameters in dimensionless form like $\eta = \mu/\sigma$, is mathematically necessary for constructing a scale-invariant statistical description of reality. This normalization renders the Fisher information metric and associated geometric quantities (distances, curvatures) invariant under global scale transformations of the underlying observables. Consequently, all physically meaningful statistical measures (e.g., SNR, CV, normalized correlation functions) become dimensionless and scale-invariant, providing the unified, observer-independent framework required by the Scale-Invariant Epistemic Framework.

This formalizes the principle that reality’s statistical structure is fundamentally captured by dimensionless ratios relative to intrinsic variability, with the standard deviation $\sigma$serving as the fundamental “ruler” for measuring fluctuations.

Q.E.D.

Standard-Deviation Normalization Proof

#Fundamental

#Given

#Goal

#Derivation

#Step

#Step

#Step

#Step

#Step

#Step 6:

#Step

#Conclusion

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